**Palindromes On The 12 hour Clock**

** **

**By: Robleh Wais**

**9/3/15**

** ****It is common knowledge that the 12-hour cycle of a clock leads to palindromic configurations predictably. I am becoming fascinated by how these palindromes occur and would like to ****explore the attributes of them. With this in mind let us look at the structure of palindromes that occur as a 12-hour clock proceeds through its cycles.**

**I will define two terms to divide the palindromes on the 12-hour clock set:**

**A full palindrome is one in which the numbers are all the same, like 1:11, 2:22, 4:44. **

**A half palindrome is one where the numbers form a palindrome by an alternating sequence of x and y variables, where x and y are integers from 0 to 9. Examples would be xyx, for ****instance, 1:21, 3:13, or 4:24. If we don't allow 0 to be considered an initial placeholder, there are only 2 four digit half palindromes, e.g. 10:01 and 12:21 for the 12-hour clock.**

**There is an ongoing challenge to discover if any pair of integers when reversed and added will eventually generate a palindrome. If we take integers larger than 2 digits many quickly ****become palindromes, while a few like 196 don't. 196 is one fundamental example of a number resisting converging to a palindrome. A term called a Lychrel number was coined **

**a Lychrel number is one, that does not converge to a palindrome after extremely long iterations of reversed addition. The problem with 196, and other integers that occur like 196, is **

**take the output and continue the process? This kind of feedback process occurs in abstract algebra all the time. ****Look at the dichotomy of palindromic types specified above. All the possible palindromic sequences for the 12-hour clock can be cataloged as an initial set. Since time-keeping on the ****12-hour clock is structured such that we won't use zero (0) as a placeholder, many palindromic sequences that could be used will not, like for instance 03:30, 01:10, 02:20, etc. Still, ****there are many possible palindromic combinations. Moreover, since we are restricting our domain to a limited set of numbers, due to the nature of addition for the 12-hour clock, we**

**have a very neat and predictable collection of palindromic numbers. But, not if we expand the derived sets to include two forms of addition. From this point on, I will abandon ****consideration of palindromic types as specified above, not relevant at present, but maybe later, since this is a living document.**

** **

**Look at any given hour on the 12-hour clock. For example, 1 o'clock to 2 o'clock. If we tabulate it, we'll find that there is a definite set of palindromes in this interval. If we do this ****for every hour on the 12-hour clock we have the following tabulation:**

** ****57 minutes are palindromic for the hours from 1 AM to the following 1 AM **

** ****The above interval is a two-hour period, and within this interval, the following palindromes occur:**

** ****2 minutes have a palindrome for the hours 10 to 12.**

** ****1 minute has a palindrome for the hour 12 am to 1 am.**

** ****54 minutes are palindromic from 1 am to 9 pm, which comes to 6 per hour (9 hours).**

**We can see that for a twelve-hour period, 57 minutes are palindromic. What is becoming much more interesting now, is the surjective sets that form, if we take the palindromic output of ****each hour starting with 1 am to the succeeding 1 am and under the operation of addition, combine them with the previous palindromic number. This would be taking the Fibonacci ****numbers of the 12-hour clock times. In parallel to this, we can add the palindromic numbers that occur hour-to-hour. Two different output sets of palindromes will be derived and ****may well have some cyclic properties. These sets are surjective mappings because the original set is being mapped to elements within it to create a new non-isomorphic set. It is a ****mapping ONTO a new set that the operation of addition (+) creates from the original set. Some algebraic sets are surjective. The natural and base 10 logarithms are surjective. So is ****modular arithmetic. It is important to note that, algebraically this is a surjective map and not an injective or bijective map. In the latter case, we are relating sets by different ****interpretations of set mapping. A bijective function simply forms an isomorphic map between elements of two sets, like 2↔B. Where the two elements are in one-to-one ****correspondence to each other. An injective map would map UNIQUE elements of one set to another via isomorphism. Please note a bijective function maps the same elements of a set ****to each other more than once without violating the notion of a function. They just have to be the same elements mapped. 9 Mod 3 =0 is a bijective map that maps 3 to itself thrice both ****ways. However, a surjective map does violate the notion of a function in that it can map the SAME element of one set to differing elements of the output set. And this is what we should ****expect on the 12-hour clock. 3:13 AM is identical to 3:13 PM numerically, so its image in a new set related by addition and would be a mapping to the same elements as on the original ****clock.**

** ****I've begun an Excel worksheet to do these operations. The worksheet so far has derived 14 additive palindromes and 11 Fibonacci palindromes. This is exciting because there is no way ****to tell where this operation will go. This sheet was simple to construct and it was manually tabulated. As mentioned above, we have a small well-ordered set (not in the ordinal set ****theory sense of that phrase) so I didn't need to write macros to get results. If I go through a 2 ^{nd} iteration of the derived palindromes are there **

**One last comment for now, I've purposely avoided going into an expansive narrative on the history of palindromic integers in mathematics. That would distract from what I consider a ****new approach to palindromic integral derivations. Those interested in the wider topic can certainly find enough material about it on the Web. If this short essay encourages that sort of ****exploration, then I'm doubly pleased.**

**You can download this file by clicking the icon below. Update: this file is now a macro-embedded file that has calculated palindromes on the 12-clock to the 3rd iteration, and ****counting...**

** Excel file for 12-Hour Clock Palindromes Palindromes on the 12 Hour Clock**

** Return to the Home Page Main Page**

__Addendum 4/16/19__

**It is important to note that the above Excel file illustrates just how random the derived palindromes are. The occurrence of a palindrome is not generated by a definable mapping of ****object sets. **

**Thus, any proof that they will occur as a result of a binary operation, on a ring of integers generated by the 12-hour clock, is not possible by the usual methods of set theory.**